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Arakelov Invariants of Riemann Surfaces

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Arakelov Invariants of Riemann Surfaces Documenta Math. 311 Arakelov Invariants of Riemann Surfaces Robin de Jong Received: August 25, 2004 Revised: June 16, 2005 Communicated by Thomas Peternell Abstract. We derive closed formulas for the Arakelov-Green func- tion and the Faltings delta-invariant of a compact Riemann surface. 2000 Mathematics Subject Classification: 14G40, 14H42, 14H55. Keywords and Phrases: Arakelov theory, delta-invariant, Green func- tion, Weierstrass points. 1. Introduction The main goal of this paper is to give closed formulas for the Arakelov-Green function G and the Faltings delta-invariant δ of a compact Riemann surface. Both G and δ are of fundamental importance in the Arakelov theory of arith- metic surfaces [2] [8] and it is a central problem in this theory to relate these difficult invariants to more accessible ones. For example, in [8] Faltings gives formulas which relate G and δ for elliptic curves directly to theta functions and to the discriminant modular form. Formulas of a similar explicit nature were derived by Bost in [3] for Riemann surfaces of genus 2. As to the case of general genus, less specific but still quite explicit formulas are known due to Bost [3] (for the Arakelov-Green function) and to Bost and Gillet-Soul´e[4] [10] (for the delta-invariant). We recall these results in Sections 2 and 4 below. In the present paper we express G and δ in terms of two new invariants S and T . Both S and T are initially defined as the norms of certain isomorphisms between line bundles, but eventually we find that they admit a very explicit description in terms of theta functions. They are intimately related to the divisor W of Weierstrass points. Of these new invariants, the T is certainly the easiest one. We are able to calculate it for hyperelliptic Riemann surfaces [13], where it is essentially a power of the Petersson norm of the discriminant modular form. The invariant S is less easy and involves a certain integral over the Riemann surface. We believe that the approach using S and T is very suitable for obtaining numerical results. An example at the end of this paper, where we compute δ and a special value of G for a certain hyperelliptic Riemann surface of genus 3, is meant to illustrate this belief. Documenta Mathematica 10 (2005) 311–329 312 Robin de Jong We start our discussion by recalling the definitions of G and δ. From now on until the end of section 4, we fix a compact Riemann surface X. Let g be its genus, which we assume to be positive. The space of holomorphic differentials 0 1 i H (X, ΩX ) carries a natural hermitian inner product (ω,η) 7→ 2 X ω ∧ η. We fix this inner product once and for all. Let {ω , . , ω } be an orthonormal basis 1 g R with respect to this inner product. We have then a fundamental (1,1)-form µ i g on X given by µ = 2g k=1 ωk ∧ ωk. It is verified immediately that the form µ does not depend on the choice of orthonormal basis, and hence is canonical. P Using this form, one defines the Arakelov-Green function G on X × X. This function gives the local intersections “at infinity” of two divisors in Arakelov theory [2]. Theorem 1.1. (Arakelov) There exists a unique function G : X × X → R≥0 satisfying the following properties for all P ∈ X: (i) the function log G(P, Q) is C∞ for Q 6= P ; (ii) we can write log G(P, Q) = log |zP (Q)| + f(Q) locally about P , where ∞ zP is a local coordinate about P and where f is C about P ; 2 (iii) we have ∂Q∂Q log G(P, Q) = 2πiµ(Q) for Q 6= P ; (iv) we have X log G(P, Q)µ(Q) = 0. Theorem 1.1 isR proved in [2]. We call the function G the Arakelov-Green function of X. We note that by an application of Stokes’ theorem one can prove the symmetry relation G(P, Q) = G(Q, P ) for any P, Q ∈ X. Importantly, the Arakelov-Green function gives rise to certain canonical met- rics on line bundles on X. First, consider line bundles of the form OX (P ) with P a point on X. Let s be the canonical generating section of OX (P ). We then define a smooth hermitian metric k · kOX (P ) on OX (P ) by putting kskOX (P )(Q) = G(P, Q) for any Q ∈ X. By property (iii) of the Arakelov- Green function, the curvature form of OX (P ) is equal to µ. Second, it is clear that the function G can be used to put a hermitian metric on the line bundle OX×X (∆X ), where ∆X is the diagonal on X × X, by putting ksk(P, Q) = G(P, Q) for the canonical generating section s of OX×X (∆X ). Restricting to the diagonal, we have a canonical adjunction isomorphism ∼ 1 1 OX×X (−∆X )|∆X −→ ΩX . We define a hermitian metric k · kAr on ΩX by insisting that this adjunction isomorphism be an isometry. It is proved in [2] 1 that this gives a smooth hermitian metric on ΩX , and that its curvature form is a multiple of µ. For the rest of the paper we shall take these metrics on OX (P ) 1 and ΩX (as well as on tensor product combinations of them) for granted and refer to them as Arakelov metrics. Next we introduce the Faltings delta-invariant. Let Hg be the generalised Siegel upper half plane of complex symmetric g × g-matrices with positive defi- nite imaginary part. Let τ ∈Hg be a period matrix associated to a symplectic g g g basis of H1(X, Z) and consider the analytic jacobian Jac(X) = C /Z + τZ associated to τ. We fix τ for the rest of our discussion. On Cg one has a t t theta function ϑ(z; τ) = n∈Zg exp(πi nτn + 2πi nz), giving rise to an ef- fective divisor Θ0 and a line bundle O(Θ0) on Jac(X). Now consider on the P Documenta Mathematica 10 (2005) 311–329 Arakelov Invariants of Riemann Surfaces 313 other hand the set Picg−1(X) of divisor classes of degree g − 1 on X. It comes with a special subset Θ given by the classes of effective divisors. A fundamental theorem of Abel-Jacobi-Riemann says that there is a canonical ∼ bijection Picg−1(X) −→ Jac(X) mapping Θ onto Θ0. As a result, we can equip Picg−1(X) with the structure of a compact complex manifold, together with a divisor Θ and a line bundle O(Θ). We fix this structure for the rest of the discussion. The function ϑ is not well-defined on Picg−1(X) or Jac(X). We can remedy this by putting kϑk(z; τ) = (det Im τ)1/4 exp(−πty(Im τ)−1y)|ϑ(z; τ)|, with y = Im z. One can check that kϑk descends to a function on Jac(X). By our ∼ identification Picg−1(X) −→ Jac(X) we obtain kϑk as a function on Picg−1(X). It can be checked that this function is independent of the choice of τ. The delta-invariant is the constant appearing in the following theorem, due to Faltings (cf. [8], p. 402). Theorem 1.2. (Faltings) There is a constant δ = δ(X) depending only on X such that the following holds. Let {ω1, . , ωg} be an orthonormal basis of 0 1 H (X, ΩX ). Let P1,...,Pg, Q be points on X with P1,...,Pg pairwise distinct. Then the formula g k det ωk(Pl)kAr kϑk(P1 + · · · + Pg − Q) = exp(−δ(X)/8) · · G(Pk, Q) k<l G(Pk,Pl) kY=1 holds. Q The definition of the delta-invariant may seem quite complicated, yet it plays an important role in Arakelov intersection theory and in the function theory of the moduli space Mg of Riemann surfaces of genus g. In fact, as has become clear from certain asymptotic results [14] [20], the function exp(−δ(X)) can be interpreted as a natural “distance” function on Mg measuring the distance to the Deligne-Mumford boundary. As to Arakelov theory, the delta-invariant plays the role of an archimedean contribution in the Noether formula for arith- metic surfaces [8] [18]. The idea that δ(X) gives a distance to the boundary is supported by this formula. The plan of this paper is as follows. In Section 2 we state a proposition and observe that it leads quickly to a formula for G. In Section 3 we prove this proposition. In Section 4 we derive our closed formula for δ. Some applica- tions of our results to Arakelov intersection theory are given in Section 5. We conclude with a numerical example in Section 6. 2. The Arakelov-Green function As was mentioned in the Introduction, the Weierstrass points of X play an important role in our approach to G and δ. The idea of considering Weierstrass points in the context of Arakelov theory is not new, cf. [6] and [14] for example. We start by recalling how we obtain the divisor of Weierstrass points using a Wronskian differential on X. Let {ψ1,...,ψg} be an (arbitrary) basis of Documenta Mathematica 10 (2005) 311–329 314 Robin de Jong 0 1 H (X, ΩX ). Let P be a point on X and let z be a local coordinate about P . Write ψk = fk(z)dz for k = 1, . , g. We have a holomorphic function 1 dl−1f W (ψ) = det k z (l − 1)! dzl−1 µ ¶1≤k,l≤g locally about P from which we build the g(g+1)/2-fold holomorphic differential ⊗g(g+1)/2 ψ˜ = Wz(ψ)(dz) .
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